March 20

Question 0

Write a=bq+r with N(r)<N(b), q,r - Gaussian integers for
a=48+27i, b=1+2i

Question 1

$5\cdot 5 = (3+4i)(3-4i)$

Why doesn't this equality contradict uniqueness of factorization to prime factors in Gaussian integers?

Question 2

Factor 65 as a product of Gaussian primes

Question 3

Use uniqueness of factorization in Gaussian integers to show that there are no integer solutions of
$x^3=y^2+1$
with even y.